Various studies have been devoted to the controllability of underactuated systems with internal controls (see [1]}, [2]}, [3]}, [4]}, [5]}). These reveal several difficulties increasing with the number of the states and with the number of distinct diffusion coefficients as a result of the notion of algebraic solvability (see [6]}). The problem of internal stabilization (via distributed control) of such systems runs deep, see for instance [7]}, [8]}, [9]}, see also [10]}. Similar complications arise in boundary controllability and stabilizability studies, see [11]}. It is also revealed that the problem of boundary controllability for the vector parabolic case is significantly more difficult than the internal one (distributed control in a part of the domain), see [12]}. In fact, an underactuated boundary control system is null controllable when the diffusion coefficients are identical (Theorem 6.1 in [11]}). However, for distinct diffusion coefficients and more than 2 coupled equations, the problem becomes more intricate. Some answers to somehow dual to stabilization problems of distributed observer design with less observations than the number of the states and in the presence of distinct diffusion coefficients have been given in [14]}, while in [15]} and in Chapter 3 of [16]}, the cases of 3 and \(m\)  linear inhomogeneous hyperbolic coupled PDEs were studied. In these works, appropriate infinite-dimensional state transformations solving operator Sylvester equations were introduced to deal with distinct elements on the diagonal of the coefficient of systems' differential operators requiring the use of higher-order spatial derivatives as measurements in order to yield Lyapunov stabilization of the observer error.
